What is the value of sin((19pi)/(12))-sin(pi/(12)) ?

The value of sin((19pi)/(12))-sin(pi/(12)) is -sqrt(3/2)

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sin((19pi)/(12))-sin(pi/(12))

Solution

sin (\frac{19 π}{12}) - sin (\frac{π}{12})

Solution

- \frac{3}{2}

Decimal Notation

- 1.22474 \dots

Solution steps

sin (\frac{19 π}{12}) - sin (\frac{π}{12})

Rewrite using trig identities:

2 sin (\frac{3 π}{4}) cos (\frac{5 π}{6})

sin (\frac{19 π}{12}) - sin (\frac{π}{12})

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 2 sin (\frac{\frac{19 π}{12} - \frac{π}{12}}{2}) cos (\frac{\frac{19 π}{12} + \frac{π}{12}}{2})

Simplify:

\frac{\frac{19 π}{12} - \frac{π}{12}}{2} = \frac{3 π}{4}

\frac{\frac{19 π}{12} - \frac{π}{12}}{2}

Combine the fractions

\frac{19 π}{12} - \frac{π}{12} : \frac{3 π}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{19 π - π}{12}

Add similar elements:

19 π - π = 18 π

= \frac{18 π}{12}

Cancel the common factor:

6

= \frac{3 π}{2}

= \frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

Simplify:

\frac{\frac{19 π}{12} + \frac{π}{12}}{2} = \frac{5 π}{6}

\frac{\frac{19 π}{12} + \frac{π}{12}}{2}

Combine the fractions

\frac{19 π}{12} + \frac{π}{12} : \frac{5 π}{3}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{19 π + π}{12}

Add similar elements:

19 π + π = 20 π

= \frac{20 π}{12}

Cancel the common factor:

4

= \frac{5 π}{3}

= \frac{\frac{5 π}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{5 π}{6}

= 2 sin (\frac{3 π}{4}) cos (\frac{5 π}{6})

= 2 sin (\frac{3 π}{4}) cos (\frac{5 π}{6})

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 π}{6})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{3}{2}

= 2 \cdot \frac{2}{2} (- \frac{3}{2})

Simplify

2 \cdot \frac{2}{2} (- \frac{3}{2}) : - \frac{3}{2}

2 \cdot \frac{2}{2} (- \frac{3}{2})

Remove parentheses:

(- a) = - a

= - 2 \cdot \frac{2}{2} \cdot \frac{3}{2}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= - \frac{2 3 \cdot 2}{2 \cdot 2}

Cancel the common factor:

2

= - \frac{2 3}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} 3}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{3}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{3}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= - \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{2}

= - \frac{3}{2}

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Frequently Asked Questions (FAQ)

What is the value of sin((19pi)/(12))-sin(pi/(12)) ?
The value of sin((19pi)/(12))-sin(pi/(12)) is -sqrt(3/2)